# Category:Successor Mapping

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This category contains results about **Successor Mapping**.

Definitions specific to this category can be found in Definitions/Successor Mapping.

Let $V$ be a basic universe.

The **successor mapping** $s$ is the mapping on $V$ defined and denoted:

- $\forall x \in V: \map s x := x \cup \set x$

where $x$ is a set in $V$.

### Peano Structure

Let $\struct {P, s, 0}$ be a Peano structure.

Then the mapping $s: P \to P$ is called the **successor mapping on $P$**.

### Successor Mapping on Natural Numbers

Let $\N$ be the set of natural numbers.

Let $s: \N \to \N$ be the mapping defined as:

- $s = \set {\tuple {x, y}: x \in \N, y = x + 1}$

Considering $\N$ defined as a Peano structure, this is seen to be an instance of a successor mapping.

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Successor Mapping"

The following 10 pages are in this category, out of 10 total.

### S

- Successor Mapping is Progressing
- Successor Mapping is Slowly Progressing
- Successor Mapping on Ordinals is Strictly Progressing
- Successor of Ordinal Smaller than Limit Ordinal is also Smaller
- Successor Set of Ordinal is Ordinal
- Successor Set of Ordinary Transitive Set is Ordinary
- Successor Set of Transitive Set is Transitive